Free subgroups in linear groups
Free subgroups in linear groups
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DOI：
10.1016/0021-8693(72)90058-0
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发表时间：
1972-02
影响因子：
0.9
通讯作者：
J. Tits
中科院分区：
文献类型：
--
作者：
J. Tits
这在特征为-7：0的域上不再成立，如有限域的无限代数扩张上的全线性群的例子所示。然而，定理2表明，这个例子在某种意义上是普遍的。rrHEOREkI 2.设V是特征距为0的向量空间，G是GL（V）的子群.因此，以下三个属性是等价的。(i)H不包含非阿贝尔自由基。(ii)G有可解正规子群R使得G，'R是局部的. fnite（即每个jinite子集生成一个$ nite子群）。(iii)G有一个有限指数子群G '，使得若V'表示k [G ']-模V的任意合成因子，k'表示V'的自同态环（即，G 'in End的扶正器，L V'），则k ′是一个积，V ′有一个k ′-基，关于这个基，表示G ′的元素的矩阵是标量乘（由k '的元素）f 0矩阵，其元素在k的素域上是代数的。
This is no longer true over a field of characteristic-7:: 0, as is shown by the example of the full linear group over an infinite algebraic extension of a finite field. However, Theorem 2 shows that this example is in some sense universal. rrHEOREkI 2. Let V be a vector space ozler aJTeld k of characteristic d# erent from 0 and let G be a subgroup of GL (V). Then, the following three properties are equivalent.(i) H contains no non-abelian free group.(ii) G has a solvable normal subgroup R such that G,‘R is locally. fnite (ie, every jinite subset generates a $ nite subgroup).(iii) G possesses a subgroup G’of finite index such that if V’denotes any composition factor of the k [G’]-module V and k’the endomorphism ring of V’(ie, the centralizer of G’in End, L V’), then k’is a jield and V’has a k’-basis with respect to which the matrices representing the elements of G’are scalar multiples (by elements of k’) fo matrices whose entries are algebraic over the prime field of k.